Is there a way to show that the following system of two equations has a solution? I don't want to find an explicit solution, but just verify its existence.
$$f''(r) + \beta \coth(r) f'(r) = \rho_0 e^{-\alpha r},$$
$$f''(r) + (n-1) \coth(r) f'(r) = C f(r),$$
where $ \rho_0 > 0 $, $\alpha > 0 $,$\beta > 0 $ and $C$ are constants.
The function $f(r)$ is a radial function (where $r$ is supposed to be the distance to a fixed point in $M$) and $f(r):M \rightarrow \mathbb{R}^+$, where $M$ is an $n$-dimensional Riemannian manifold, complete and simply connected, with negative constant sectional curvature.
